L(s) = 1 | + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (−1.21 − 0.825i)11-s + (1.78 − 0.858i)13-s + (−0.988 + 0.149i)14-s + (0.955 + 0.294i)16-s + (−0.5 − 0.866i)18-s + (−0.955 + 1.65i)19-s + (0.623 + 0.781i)20-s + (−0.914 + 1.14i)22-s + (−0.365 − 0.930i)23-s + ⋯ |
L(s) = 1 | + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (−1.21 − 0.825i)11-s + (1.78 − 0.858i)13-s + (−0.988 + 0.149i)14-s + (0.955 + 0.294i)16-s + (−0.5 − 0.866i)18-s + (−0.955 + 1.65i)19-s + (0.623 + 0.781i)20-s + (−0.914 + 1.14i)22-s + (−0.365 − 0.930i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7421966515\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7421966515\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0747 + 0.997i)T \) |
| 5 | \( 1 + (0.733 + 0.680i)T \) |
| 7 | \( 1 + (0.222 + 0.974i)T \) |
good | 3 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 11 | \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 19 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \) |
| 53 | \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-1.03 + 0.702i)T + (0.365 - 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.810375268671154137840887833694, −8.282780961784070629661467792034, −7.78784869548067765973233771205, −6.41901768692884608512538168895, −5.56977625187679975167025913557, −4.52857463467434597721411774010, −3.67213409052099819372134629921, −3.41186134193345065387762718787, −1.59204853405602646074165138011, −0.55375238537567859369689011395,
2.05444645754324081209956134039, 3.31261501664084794861543047146, 4.29035924587682725835404829352, 4.97826577629306245220804100387, 5.99332177436876567915482297065, 6.79395947623489539476061059657, 7.31544466653034638617719848251, 8.187368335387522098140219097621, 8.777603992480431372493546584994, 9.588148468465758752830043369652